Energy Pressed into the Corners: The Proton's Core Mass Radius from the Consolidation Law of the Tetrahedral-Octahedral Lattice, r_p/√3 ≈ 0.485 fm

In the Theory of Space (TOS) program the proton is the octahedral cavity of level 66 of the doubling ladder of the tetrahedral-octahedral lattice (TOT): an octahedral void enclosed by eight tetrahedral cells of the same level, with octaradius equal to the proton charge radius rp, and with mass given by the closed-form relation mpc² = 4ħc/rp (Cpr = 4) [2]. We first remove an apparent conflict between two mass laws — mass ∝ 1/size between levels of the ladder and mass ∝ volume within one cell — by noting that both follow from a single mean energy density ρ̄ ∝ ħc/r⁴; for the octahedron it is exact, ρ̄p = 3ħc/rp⁴ ≈ 1185 MeV/fm³, and gives an MIT bag constant B1/4 = 218 MeV. We then ask where inside the cavity the energy resides. The consolidation law of the lattice divides every octahedron into six corner octahedra and eight face tetrahedra of half the edge. If the energy is pressed away from the tetrahedral walls into the octahedral corners — the narrow crevices of the cavity wedged between the surrounding tetrahedra — and this rule is applied recursively, the root-mean-square radius of the energy distribution is, with no free parameter, rm(n) = rp·√(1/3 − 4⁻ⁿ/30), i.e. 0.4605 fm for a uniform cell (n = 0), 0.4793 fm at the first level of consolidation and 0.4854 fm = rp/√3 in the limit. The result is insensitive to the depth: every n ≥ 1 lies in 0.479–0.485 fm. An independent construction gives the same number: if the cavity is read as a hollow shell — the octahedron of level 66 with its central node star (an octahedron of level 65, 1/8 of the volume) left empty — a uniform energy density in the shell yields rm = rp·√((3/10)(31/28)) = 0.4845 fm. It agrees with the nucleon core radius 0.48 ± 0.01 fm and the gluonic mass-core radius 0.51 ± 0.05 fm summarized in [8], and it marks extractions of the full mass radius above ≈ 0.56 fm as necessarily including mass outside the cell (the meson cloud). In the limit the energy lies uniformly on the three axes of the octahedron. We state an explicit falsification criterion. Version note: attribution of the relation r_p = 4ħ/(m_p c) corrected — first written down by Haramein (2012), derived independently by Trinhammer and Bohr (2019); in this series ƛ_p and r_p are the octaradii of lattice levels 64 and 66. Results unchanged. Contact: stakhoff@gmail.com · https://theoryof.space